For fixed \((t,s)\in [0,1]\times [0,S]\) , define \(\varphi (\theta ,t,s)\)  as the distribution of \(\theta _\phi (s;t)\) . Using the classical stochastic theory [1]}, \(\varphi (\theta ,t,s)\)  is a solution to (REF ). For fixed \((t,s)\in [0,1]\times [0,S]\) , we prove (REF ) due to (REF ). Finally, using (REF ) and (REF ), we obtain that
\(\lim _{(t,s)\rightarrow (t_0,s_0)}W_2(\varphi (\cdot ,t,s),\varphi (\cdot ,t_0,s_0))\le \lim _{(t,s)}\left(\mathbb {E}\left(\left|\theta _\phi (s;t)-\theta _\phi (s_0;t_0)\right|^2\right)\right)^{1/2}=0\,,\) 
